The report is devoted to the results of the numerical study of the virtual cathode formation conditions in the relativistic electron beam under the influence of the self-magnetic and external axial magnetic fields. The azimuthal instability of the relativistic electron beam leading to the formation of the vortex electron structure in the system was found out. This instability is determined by the influence of the self-magnetic fields of the relativistic electron beam and it leads to the decrease of the critical value of the electron beam current (current when the non-stationary virtual cathode is formed in the drift space). The typical dependencies of the critical current on the external uniform magnetic field value were discovered. The effect of the beam thickness on the virtual cathode formation conditions was also analyzed.

Synchronization processes in networks of spatially extended dynamical systems are analytically and numerically studied. We focus on the relevant case of networks whose elements (or nodes) are spatially extended dynamical systems, with the nodes being connected with each other by scalar signals. The stability of the synchronous spatio-temporal state for a generic network is analytically assessed by means of an extension of the master stability function approach. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. The efficiency and reliability of this method is illustrated numerically with networks of beam-plasma chaotic systems (Pierce diodes). We discuss also how the revealed regularities are expected to take place in other relevant physical and biological circumstances.

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The task of planning an experiment can be divided into two stages: screening experiments conducting, step-by-step experiments performed on complete factorial design or on fractional replicates. As the number of publications on the practical use of plans-experiments in construction is insufficient, the paper presents appropriate recommendations in common form. Screening experiments are usually conducted with the aim of approximate orientation among many factors and for more precise future planning. These experiments can be carried out on the minimum fractional replicates making it possible to obtain the first order model. After obtaining the model, the factors that are included in it are appropriately ranked. Using the results of ranking, researchers take into account economic considerations, leave away unimportant factors and conduct experiments with remaining factors to obtain a mathematical model of the process. In order to find out the nomenclature of these factors, screening experiments are carried out. These experiments are performed to obtain an approximate dependence between the outcome of the process and the factors. This dependence is used to determine the influence that each of the factors has on the experience result. When the crucial factors are considered, it is possible to plan the basic experiment.